The most problematic aspect of Gmail Android app for me is it's poor offline management capabilities. For example, I can choose what labels I want to synchronize, but the synchronization period is specified globally, for all labels.
I'd like to say for which label, how long I want to save something offline, I'd like to be able to pin something to stay in offline cache forever, or to set cache size constraint (instead of time constraint).
Is there a free Android app offering such features?
No. As of now, there doesn't seem to be any app that can selectively synchronize different labels for your Gmail account.
Actually, if I need to have something to be always available offline, I would just clip it to my note-taking application on my phone (such as Evernote or Google Keep), which will also be syncing the data to the cloud so that the data is accessible even if my phone is having some problem. And, if the data is something sensitive, then you shouldn't actually be making it easily accessible, bringing it offline in your Gmail app.
Since most of the mobile devices these days are almost always online, I don't think that we will be seeing such a feature in any app anytime soon, unless a public demand for the same rises (which, I don't think will be happening in the current scenario).
